you can use strongly typed view and return the model from the btn click post event.
for example:-
[HttpPost]
public ActionResult Save(Student sdtudentModel)
{
return View(studentModel)
}
Then in View you can use the code to bind the dropdown List
@Html.DropDownListFor(model => model.StudentID, Model.StudentsList)
StudentID will be the property that will store the selected Value
StudentIDList will be IEnumerable<SelectListItem> that will store all the dropdown values.
Both of these will the properties in your Model.
By this dropdown will retain value on postback(till the returned model has value for StudentID)